if x = asecA+btanA and y= atanA+bsecA . prove that x^2 - y^2 = a^2 - b^2

Given, x = asecA+btanA and y= atanA+bsecA

Now, 

LHS = x^2 - y^2

= (asecA+btanA)^2 - (atanA+bsecA)^2

= a^2sec^2A + b^2tan^2A + 2absecAtanA - a^2tan^2A - b^2sec^2A - 2abtanAsecA

= a^2 (sec^2A - tan^2A) + b^2 (tan^2A - sec^2A)

= a^2 (1) + b^2 (-1)                (Since, 1+tan^2A = sec^2A)

= a^2 - b^2

= RHS